Problem 9 - Algebra and Trigonometry

Question

Express the given quantity in terms of the indicated variable. The distance (in mi) that a car travels in \(45 \mathrm{min} ; \quad s=\) speed of the \(\operatorname{car}(\text { in } \mathrm{mi} / \mathrm{h})\)

Step-by-Step Solution

Answer

Distance = \(\frac{3}{4} s\).

Step 1: Understanding the Problem

We are asked to find the distance a car travels in 45 minutes and express it in terms of the speed of the car (s). The speed is given in miles per hour (mi/h), and we need to relate this to distance.

Step 2: Convert Time Units

Since the speed is given in miles per hour, we need the time to be in hours as well. Convert 45 minutes to hours: 45 minutes is equivalent to \(\frac{45}{60}\) hours, which simplifies to \(\frac{3}{4}\) hours.

Step 3: Apply the Distance Formula

The formula for distance is \( \text{Distance} = \text{Speed} \times \text{Time} \). Here, time is \(\frac{3}{4}\) hours and speed is given as \(s\) miles per hour. Thus, the distance can be written as \(\frac{3}{4} s\).

Step 4: Express in Terms of the Indicated Variable

We already have the distance expressed in terms of \(s\), so the distance in miles that a car travels in 45 minutes is \(\frac{3}{4} s\). This expresses the quantity in terms of the indicated variable.

Key Concepts

Converting Units

Converting units is a crucial skill in mathematics, especially when working with formulas involving speed, time, and distance. In the given problem, the time is initially provided in minutes, yet the speed is in miles per hour. This difference means we need to convert minutes to a consistent unit—hours—before using these values in the distance formula.

To convert minutes to hours, remember that there are 60 minutes in an hour. For example, with 45 minutes, the conversion to hours is done by dividing by 60: \[\text{Time in hours} = \frac{45}{60} = \frac{3}{4} \text{ hours}\]This conversion is important because it ensures both the speed and time units are consistent, allowing calculations to be accurate when applying formulas. Keep in mind:

• Always check the units of each quantity before performing any calculations.
• Use fractions to represent converted units to avoid rounding errors during intermediate steps.

Speed

Speed is defined as how fast an object is moving, usually expressed as distance traveled over a certain period of time. In this exercise, speed is given in miles per hour (mi/h). When dealing with speed, always consider the unit of time it represents, as it directly affects calculations.

Given the speed \(s\) in the exercise, it indicates the distance the car travels every hour. This unit consistency makes it simpler to apply the distance formula. In many real-world applications, understanding speed helps determine how long a journey will take or how far it can go within a set time. Key takeaways:

• Speed is a scalar quantity, meaning it only has magnitude, not direction.
• Ensure speed units align with the time units before calculating distance or time.

Time

Time plays a vital role in calculations involving speed and distance. In mathematics problems like the one at hand, time generally has to be consistent across all elements of the formula. Originally given in minutes, learning how to convert it accurately is key to success.

After conversion, the time unit changes to hours, which aligns perfectly with the speed unit (miles per hour). Applying the distance formula becomes straightforward when every component is compatible:\[\text{Distance} = \text{Speed} \times \text{Time} = s \times \frac{3}{4}\]Thus, the time of \(\frac{3}{4}\) hour gives a clear and simple multiplication step.

• Time should be expressed in the same unit as used in other related quantities.
• Reliable conversions ensure formulas remain accurate and understandable.